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Analysis of Simultaneously Measured
Pressure and Sandface Flow Rate in
@Transient Well TestingF. Ku?uk, SPE, Schl.mbc%er Well Services
L. Ayestaran, SPE, Schl.mberger Well Services
SUrmnary
New well test interpretation methods are presented that
eliinate wellbore storage (aftertlow) effects. These new
methods use simultaneously measured sandfuce flow r“te
and weffbore pr~ure Ma. It is shown that formation
behavior without storage effects (unit response or in-
fluence function) can be obtained from deconvolution of
Jkmdface flow rate and wellbore pressure data. The
storage-free formation behavior canbc analyzed to iden-
ti& the system (reservoir flow pattern) that is under testing
and to 6stimate its parameters. Convolution (radial multi-
rite) methods for reservoir parameter estimation and a
few synthetic examples for deconvolution and convolu-
tion also are presented.
Introduction
Welj testing with measured sandface flow rate can be
traced to tbe beginning of reservoir engineering. The rate
must + measured over time to calcuk and{or approx-
imate constant rate to obtain even a single reservoir
parameter from pressure measurements. Tfds approiimatc
.comstant rate has been sufficient for estimatingptmneabili-
ty, skin, and initial formation pressure during the ra&d
infinite-acting period. During tbk period, the well should
produce at? conkant rate at the sandface or at a zero rate
if a buildup test is condu~ed. Because of compressible
fluid in the production string (weflbore storage effects),
it (alces a long @e to reach the radial infinite-acting
period. The effect of outer boundaries also may stat
before the end of the wellbore storage effects.
In gener~, the storage ~pacity of the wellbore,
wellbore geometry, rim-wellbore complexities, and ex-
ternal boundaries affeqt transient bebavior of a well. Dur-
ing the. aualysis of pressure-time data, each of these
phenomena and its dtition must be recognized for the
aPPEcatiOn Of SeMilog and typecurve techniques to deter-mme formation flow capacity (k/z), @mage skin, and
average formation pressure. The influence of these
phenomena on transient behavior of a wg?fl progresses over
time. For the sake of convenience, the test time can be
divided into ~ periods according to which phenomenon
is affecting the pxessuze. These periods are defined as
follows ,
EarIy-T@ Period. The combmed effects of wellbme
storage, +nmge skin, and pseudoskin (which include par-
mwmt f9.33 sww .f P.t,.h+ EWW.IS
REBRU.4RY 1985
5R!5 lal’71
tkd. penetration, perforation, aciduing, fractures, nOn-
Darcy flow, and permeability reduction caused by gas
saturation around the wellbore) dominate pressure
behavior. The stratification and dud porosity also may
affect wellbore pressure during this period.
Middfe-Time Period. During thii period, radial flow is
es@blished. Conventionally, semilog techniques are used
to determine formation kh and initial pressure and skin.
Late-Time Period. During tbk period, outer boundary
effects start to distort the semilog straight line. For ex-
ampIe, the gas cap shows a curve-flattening effect on log-
log and Homer plots.
Sometimes the separation of these periods from each
otier is impossible; particularly, the effects of bottom-
water influx andtor gas cap may start during the. middle-
time period. Thus, the semilog approach sometimes can-
not be app~,ed at all.
Furtbeirnore, the drawdown or buildup tests as con-
ducted today tend to homogenize tie reservoir behavior.
In other words, ” most of the reservoirs behave
homogeneously during the storage-free radial intinite-
acting period because most of the heterogeneous behavior
takes place during the early-time period.
The &pe-cume approaches have been introduced to
overcome some of these problems. The theories, applica-
tions, and elaborations of the type-curve methods, as welf
as WY” references, can be found in Ref. 1. In 1979,
Gringarten ez al. 2 introduced new type-curves that use
different pammetetiation than the earlier ones, namely
Rhmey, 3 Agar.val et al., 4 McKinley, 5 and Eafkmgher
and Kemch6 types. All the type curves presented by these
au~ors, and many others, were developed under tie
assumption that ~e fluid compressibility (density) in the
tubing and anmdus remains constant during @e test period.
During the early time, pamicularly for buildup tests, shut-
in pressure incrkases ve~ rapidly; thus, the compresiibJl-
ty is usually higher than the compressibility of the fluid
in the reservoir for producing wells. Since the piessure
in the wellbore is a fgnction of the depti, the com-
pr&ibfity of the fluid at the wellhead can be 10 or even
100 tink geater than the compressibility of the fluid at
the @ttOm. T@, the assumption that the ‘wellbore storage
coefficient is constant during the drawdown, and pal-
ticukrly during buildup, may not be correct. A variable
weflbore storage coefficient alone makes the application
323
of type-curve methods afmost impossible. The combina-
tion of variable or even constant wellbore storage with
wellbore “geometry timber complicates the type-curve
~tching process. Moreover, wellbore pre.wure &ta afone
tiY not indicate changing wellbore storage.
Some of the problems inherent with the use of type-
cume methods can be efiiated by the sbnuftaneous use
of measured sandface flow rate and pressure data.
‘fhe purpose of this work is to study the use of the
measured sandface flow rate in a broad sense with regard
@ transient weU”testing. Furthermore, we explore the use
of convolution and deconvolution $1 the interpretation of
pressure beliavior of a well with afterflow (buildup case)
or wellbore storage (drawdown case).
Background
The use of tie sandface flow rate in transient testing is
not riew. To oti knowledge, van Everdingens and
Hurst7 were the first to estimate and use the sandface
flow rate to calc~ate the wellbore pressure. To do this,
they approximate the stidface flow rate by the forr+nda
qsf=(’-’-”o$. . . . . . . . . . . . . . . . . . . . . ...(1)
where 13 is a positive constant. These authors stated that
the constant, ~, cm be determined from well aad res?r-
voir parameters. Using tie above formula and the con-
volution integd, van Everdingen7 and Hursts presented
an expression for the weUbore pressure with a variable
wellbore storage effect. Gladfefter et al. g presented’ a
method to determine the formation kh from pressure and
aftertlow ikita. The afterflow &ta were obtained by
measuring the rise in the liquid level in the ivellbore.
Ramey 10 applied the Gladfelter approach to gas’ well
buiklup tests.
A considerable amount of work afso has been done on
multirate (vimiible) rate tests dining the last 30 yeys.
However, these are basically sequential constant-rate
dratidowns; only transient pressure is measured and rate
is assuined. constant during each drawdown test. The
techniques related to thk type of multirate tests afso can
be found in Ref. 1. AU the ivork mentioned so far de#s
with the direct problem. In other words, the constant-rote
solution (the influence or the unit response function) is
convolved (superimposed) with the the-dependent inner
@~dary condition to obtain solutiom to the diffusivity
equation. This process is calfed “convolution.”
Hutchison and .%kora, 11 Katz et af., 12 and Coates er
af. 13 presented methods for determ.inin g the influence
function ~ectly from field data for aquifers. The proc-
ew of determining g fie influence function is cafled’ ‘decon-
volition. ”
Jargon and van Poolen 14 were perhaps the first to use
the deconvolution of variable rate and pressure data to
compute the constant-rate pressure behavior (the inffnence
fiction) of the formation’in well testing. Bostic et al. 15
used a dwonvolution tecbni@e to obtain a constant-rate
solution from a variable rate history with a known pressare
history. They also extended ~e deconvolution technique
to combine production and buldup data as a single test.
Pascal 1s also used deconvolution techniques to obtain a
constant-rote solution from variable rate (measured at the
surface) and pressure measurements of a drawdown test.
More recently, Metier et al. 17 have used sandface
flow measurements with pressure data fo? buildup test
amafysis. This has been the first successful attempt to use
direct measurements of sandface flow rate data in weU
testing. They showed that the Homer method can be
modilied to ‘&ach a semifog stmight line earlier km type-
curves or the 1 I%-cycle @e irdcates.
Theoretical Developments
During the last 4 or 5 decades, many solutions have been
developed for transierit fluid, flow through porous media.
me superposition theorem (Dahamel’s tieorem) has been
used @derive solutions for time-dependent boundary con-
ditions from time-independent boundary conditions. For
example, the multiple-rate testing is a special application
of the superposition theorem.
fn their classic paper on unsteady-state flow problems,
van Everdingen ~d Hurst’s presented the dimensionless
wellbore pressure for a continuously varying flow rate as
PwD(b)=mJ(f9Pso (b)
+ jtDqD‘(T)p@ (f~ - ~)dr. . .
0“
(2a)
An alternative form to E.q. 2a can be obtained by an in-
tegration by parts as
PwD(tD) ‘qD(tD)psD@)
+~fZD(7h’D&r~)dC . . .. . . . . . ..(zb)
0
where
pwD = *1%-PM$l!
0.0002637kt~D =
qipctr$
psD = PD+L$,
PD(tD) = the dimensiodws sandface pressure for
the constant-rate case without wellbore
storage and sf& effects,
S = steady-state skin factor,
qD (tD) = !hf(tD)fq, ,
qi(b) = dqD(tD)/dtD ,
q, = reference flow rate—if the stabilized
constant rate is available, then q,
should be replaced by q~,
q~t) = variable sandface flow rate (flowmeter
readings) , and
qD(2Li) = dheUSiOdeSS sandface rate.
324 JOURNAL OF PETROLEUM TECHNOLOGY
1.0.
shut-m Time, AI, hrs
Fig. I—Dimensionless sandface flow rates for constant
tiel Ibore storage and exponential decfi ne cases.
Although traditionslfy the skin effect is considered a
dimensionless quantity different from the dimensionless
formation pressure, the skin effect will be treated here
as part of the imer boundmy condition for the solution
of a unit rate production case. This boundsry condition
is known as the homogeneous boundary condition of the
thiid kind..
It shoufd be emphasized that Eqs. 2a and 2b csm be ap-
plied for many reservoir engineering problems. The
linesrity of the diffusivity equation allows us to use Eqs.
2a and 2b for fractored, layered, anisotropic, snd
heterogeneous systems as long as the fluid in the reser-
voir is single phase. F@. 2a and 2b cm be applied to both
dmwdown and buifdup tests if the initial conditions sre
known. For a reservoir with m idkd constant snd uni-
form pressure distribution [pD(0)=O], Eqs. 2a and 2b
can be expressed as
‘D
PWD(fD)=f q~(7)p@(tD–T)dT . . . . . . . . . . . .(3a)
0
=SgD(tD)+ fDqD(T)p,’D(tD –~)dr. (3b)
0
Furthermore, Eq. 3a sfso can be expressed as
PWD(tD)= ~’DqA(iD–T)P@(r)dT. . . .’ . . . . . . . . (3c)
.0
fn Eqs: 3a tid 3c, it is assumed that q~(rD) exists. If
qD(2D) is constant, then Eq. 3b must be used. ls@. 3a
snd 3C we known as a Volterra integral equation of the
first kind aod the convolution type.
Akhough it is assumed that PO is a constant-rate solu-
tion without stomge effect, mathematically and physical-
ly it can be a solution of a constant-storage case. In
pmctice, p$D afways will be affecti by the welfbore fluid
that occupies the volume below the flowmeter u,!dess the
ssndface rate is measured through perforations. However,
the volume below the flowmeter will lx sms.lf for most
wells, since the flowmeter and the pressure gauge usual-
ly csn be placed just above perforations. Therefore,
throughout this paper, we will assume that p,D is not af-
fected by the fluid volume below the flowmeter.
FEBRUARY 1985 ““
TABLE l—HOMOGENEOUS RESERVOIR ROCK
AND FLUID DATA
8, bbl/STS0,, psi-’
ccl
h. ft
p, Cp
.6
1.0,..5
1,000
100
40
2,666,94
3,000
0.35
10,000
0.76 x10-4
0.8
0.2
The purpose of the test welf interpretation, as stated by
Gringarten et al., z is to identify the system and deter-
mine its goverrdng psmrneters from measured dsts in the
wellbore and at the wellhead. Thk problem is known as
tie inverse problem. Tbe solution of the inverse problem
usuafly is not unique. As Gringarten et al. 2 pointed out,
if the number snd tie range of measurements increase,
the nonuniqueness of the invers.k problem will be reduced.
Thus, combining sandface flow rate with pressure
measurement will enhance the conventional (including
type-curve) weU test interpretation methods.
AS m inverse problem, the ssndface pressure, psD(tD),
hss to be determined by the deconvolution of the integral
in Eqs. 3a, 3b, and 3c. As stated previously, p,D(tD) is
the solution for the constant-flow-rate (sandface) case.
Taking the Lsplace transform of Eq. 3a and solving for
~@(s)* yields
~~D ($)= ~:D ‘s)=, . . . . . ..8. . . . . . . . . . . . . . . . . . (4)
where s is the Laplace transform variable.
The Laplace Esnsform of P,D in Eqs. 3b and 3C will
be the same m Eq. 4, keeping in mind that p,D(o) =S.
Thu3, we hzve only one operational form of the convolu-
tion intograf given by Eqs. 3a, 3b and 3c. The superposi-
tion thmrem io this case is notliig more thsn the
convolution of P,D (tD) aod qD (tD). The Laplace tmns-
form of the convolution intepal allows us to express the
convolution integral in msny different forms. Further-
more, the kemef solution csn be a solution of constant-
rate or comtsnt-pre-wure case for the convolution integral.
If the wellbore storage is constant, the dmensiordes:
sandface flow rste can be expressed as4,1s
dp ~,D (tD)q~(tD)=l–c~—, (5). . . . . . . . . . .. . . . . .
dtD
where
qD(tD) ‘q,f(t)fqr.
As a special application of Eq. 4, the dimensionless
wellbore pressure solution for the constant-storage case
cao be written directly from Eqs. 4 and 5 as
~sD ($)
~wD (s)=
1+ CDS2F,D(S)’ ““””’”’””””””’”””””
(6)
Vhmughout lhls paper, the runclian ?(.9 w be Mod the Wlace tram form.! tha
f.ncsion F(t).
325
/’-...O.W9
,,,.”.., !.!
k&,~.. ,~.z , ~.,10“
shut-to n.., M, hr.
Fig. 2—Shut-in pressures for constant wellbore storage and
exponential decline cases.
where ~$D (s) is the dimemionIess sandface pressure for
the constant-rate case witbout storage effect but including
skin.
Van Everdingen and Hurst 18 presented an equatiOn
simifa to Eq. 6, and Aganval et al. 4 presented the same
equation as an integrodifferentid form for radial systems.
Cmco-Ley and Mmaniego 19 {for fractured reservoirs),
and Kufmk and Kirwan20 (for partially penetrated wells)
presented the same expression for the dimensionless
weflbore pressure as in Eq. 6.
on the other hand, the dimensionless sandface flOW raE
can be obtained directly from Eqs. 5 and 6 in terms of
the dimensionless formation pressure as
1~~(s)= . . . . . . . . . . . . . (7)
St 1 + CDS2ESD (s)1
Fig. 1 presents vafues of qD wdculated from Eqs. 1 and
7 as a function of real time for a buildup test using reserv-
oir and fluid properties given in Table 1. As can be seen
tlom this figure, for the exponential&clime case, the samd-
face tkw rate declines faster than the constant-wellbore-
storage cwe. IZamey and Agarv.m121 also pcesentd values
of qD(tD) as a function of tD for various skin and storage
constants:
Another important application of the convolution in-
tegral given inF.qs.’3a and 3C was resented by van Ever-
dingen, 7 Hurst, g and Ramey*J for calculating the
wellbore pressure by using Ea.. 1 and the line source
solution.
For a finite wellbore radius, tie dimensionless wellbore
pressure solution afso can be written directly from Eqs.
1 and 4 for the exponential sandface rate decline case as
~wD(.,=y —. ,., ,.,. .,, ,,, ,. .,, . . . . . . . . (8)
‘fhe exponential constant, i3, is giVeII M V~ Eve@gen7
as
(3=m#yic,rw210.000264 k,
where u can be determined from welfbore pressure data,
such as the wellbore storage constant. Note that (3 is a
dimensionless constant like CD.
326
As Ramey 10 noted, i3 cannot be estimated as re@ily
as the weflbore storage constant, CD. However, in prin-
ciple, Eq. 9 has a much more ~ediate cO~~tiOn wi~
the real systems. In fact, the pressure increwes more
rapidly during the early-time buildup tests; then it slows
down during the transition period and boilds up vay sfow-
ly during the semilog period. Because of the rapid change
of pressure, the wellbore storage wilf decrease condnuoLls-
Iy except in the case of phase redistribution.
In many cases, it is difficult to recognize changing
wellbore storage effdcts because it is a gradual and con-
tinuous change. Furthermore, tie fifi~ clOsing time Of
the welffread valve also will affect pressure at tie same
time.
Most of the work thus Em in early-time analysis has been
directed toward the construction of type curves from the
solutions of Eq. 7 for a constant wellbore storage for dif-
ferent wellbore geometries, such as fractured wells, par-
tial penetration, etc. Type curves for Eq. 8 for different
vafues of O and skin also can be developed and used for
graduafly decreasing wellbore storage cases to determine
skin snd W?.
Fig. 2 presents a semilog plot of p ~, (At) vs. At,
calctiated from Eqs. 7 and 8 by using reservoir and fluid
parameters given in Table 1. A8 shown by thk figure,
tle exponential decline case approaches a semilog straight
line earlier than the constant-wellbore-storage case.
The most important point to be made frO.m tbe abO~e
discussion and buildup data prc%ented in Fig. 2 is that the
principle limitation of the type-curve analysis stems from
the lack of information about the sandface flow rate
behavior. Thus, the type-mu-w analysis usuafly is used
for quabtive answers and supported by the semilog
analysis. For quantitative analyses of the early-time data,
it is necessary to measure the sandface flow rate. Fur-
thermore, the use of measured sandface flow rate also can
improve the semilog anzfysis. In the following s&tiOns,
the convolution and deconvolution of simultaneously
measured sandface flow rate and pressure data will be
shown to obtain the formation pressure and parameters.
Convolution (Superposition)
Continuous Multirate Method. The simplest approach
to solving the convolution integral is to assume a psD
function in Eq. 3a. This psD tb~On cO~d ~ a line-
source solution, infinite conductivity vertical fractured
solution, etc., for the constant-flow-rate case or the cOn-
stant-pressure case. The chosen pa function can be con-
volved with qD (sandface flow rate) by using the
convolution integril (Eqs. 3a tkougb 3c) to modify the
time function or pwD. For example, the cOnvOlutiOn
(superposition) of variable rate with the log approxima-
tion for tbep,D function and wellbore press~e CCJMMOR-
ly is used for the anafysis of muldrate tests. The same
technique also can be used for the analysis of buildup or
drawdown tests with measured sandface flow rate data.
USing the 10g approtition for P.D in f%. 3a! fie
change in measured pressure in oilfield units can be ex-
pressed as
Af)&t) =m@(,)@og(t-r) +~]d~, . . . . . . . . (9)”
0
JOURNAL OF PETROLEUM TECHNOLOGY
where
Ap ~f) ‘pi ‘p tit),
‘=”’’’+’”g(*)-’’’”
and
162.6 @q
~.—
kh
Eq. 9 can be rewritten as
where b= ,~m.
Letusapproximate the integral in Eq. 9 by the Riemamr
sum, which yields
AP.jc(/n) =
qD (tn)
where t. is the measured (discrete) time point. This equa-
tion has been presented elsewhere for multiple-rate
analysis. Eq. 10 gives the condnuous form of the muhirate
(variable-rate) equation. Any integration techniques, as
well as the Riemarm sum used in Eq. 11, can be used to
evaluate the integral given in Eq. 10. A plot of the left
side vs. the first term of the right side of Eqs. 10 or 11
will yield a straight line with a slope m and an intercept
b. For buildup tests, qD(f) should be replaced by
1 –qD (At) and AP ~f(At) by Ap ~, (At) =p ~, (At)
–p~At=O) in Eqs. 9 through 11.
The major advantage of continuous variable-rate test
(rate measure just above the perforation) over the con-
vention multirate testis tit the wellbore storage effects
are minimized. The wellbore storage effect has not been
discussed in the literature for the conventional mtddrate
tests. The wellbore measured pressure used in the con-
ventional malysis should be taken from the storage-free
in ftite-acting period. Thus, the end of the storage effect
should be determined. In other words, the sandface rate
should be equal to the constant-surface rate for each
drawdown or buildup period.
The second problem with the conven!iond multirate test
is the surface step rate change cannot be taken as a step
rate change for the sandface. Neglecting the continuous
rate change from one rate to another will affect the result
of the conventional multirate test analysis.
PEBRUARY 1985
As noted earlier, the continuous variable-rate test sug-
gested here will not eliminate completely the effect of the
wellbore storage, since there is a finite volume between
the bottom of the well and the flow meter, but it will
minimize the wellbore storage effect.
The method suggested by Gladfelter et al. 9 further
simplifies Eq. 12. However, it does not improve the
muhirate method. The disadvantage of the Gladfelter et
al. 9 method over the mukirate method is that here is a
pxsibfity of the existence of one to three different straight
lines.
Motiled Homer Method. Using the measured sandface
rate &ta, Meunier et al. 17 presented a modification of
the Homer time ratio, which they named “the rate-
convolved buildup time function. ” They showed that the
time required for the start of the semilog straight line can
be reduced consi&rably by using the rate-convolved
buildup time function instead of of the conventional
Homer time ratio. Metier et al. 17 gave detailed ex-
planations on how to modify the Homer dme ratio if the
sandface rate measurements are available. In this section,
the fundamental mture of tie Homer and modified Horner
methods will be examined.
The buildup test is perhaps the most popular transient
testing practiced by the oil industry over tie last 3 decades.
The Horner method used for the analysis of the buildup
test is appealiig for its simplicity, generality, and ease
of application. The reliability of kh, skin, and the ex-
trapolated pressure estimated from the Homer method
depends on the slope of the Homer semilog straight line.
The assumption required for the Homer method is that
the sandface flow rate becomes zero during the semilog
period. Flom a theoretical point of view, as long as the
measured pressure increases at the wellbore, the sand-
face rate never will be zero unless the fluid in the wellbore
is incompressible. Thus, rhe effect of the decaying sand-
face flow rate on the Horner semilog straight line wifI
be investigated in this section.
The convolution integral given by Eq. 3b can be writ-
ten for buildup tests as
PDS(~pD +AtD)=%D(ArD) +PD(t@ +AtD)
-~[1-qD(T)@;(MD-~)d~, . . . . . . . ..(12)
o
where
and q is the constant rate. before shut-in.
For tie sake of simplicity, let us assume that after shut-
in, the afterflow rate decliies exponentially, as suggested
by van Everdingen6 and Hurst7 so that Eq. 1 becomes
qD(At) =e ‘-&, (13)
where a is determined from measured sandface flow rate
data. For e~ple, if sandface rate measurements are
available, a in Eq. 13 can be determined by the least-
square.s curve fitting of Eq. 13 to the sandface rate data
327
Horner & Modified Horner Time Functions
Fig. 3-Modified Horner and Horner plots for exponential
dec~ne sandface rate,
Fig. 4—Functions for the start of the Horner and modified
Horner semilog straight tines.
Substitution of the exponential integral solution forp~
and Eq. 13 for qD in Eq. 12 (the details of the deriva-
tion are given in Appendix A) yields
pi-pw’’’)=~[’”g(%)+al (14)The constant a in Eq. 14 can be considered an afterflow
parameter. The term l/2.303cYAt will modify the Horner
&me ratio, (tP +At)/At. A semilog plot of pw~ (At) vs. An-
tilog {log[(tP +At)/&] + l/2.303aAt} wilJ yield a ~ght
liie with a correct slope. This straight line starts much
earlier than the Homer “setnilog straight line, as shown
in Fig. 3 (approximately one-half cycle earlier). In fact,
Chen and Brighamz found that the correct semilog
straight liie on the Homer plot is obtained after an ex-
Fig. 5—OimenslonIess times for the start of the Horner and
modified Horner semilog straight lines as a function
of 0.
tremely long time period. It is obvious from Eq. 14 that
as At increases, 1/2. 303cYAi becomes smaller, but log
[[tP +Ar)/At] decreases as well. Thus, the Homer plot,
as it is seen in Fig. 3, asymptotically approaches the cor-
rect semilog straight line.
If the term (ZP +Ar)/At is very large compaed to the
term l/2.303aAt, then the Horner and modified Homer
straight lines are almost identical at large At. In other
words, if tp is veiy large compaed to the maximum shut-
in time, then the correction caused by the aftertlow
becomes almost negligible as At becomes large.
Before determining g an approximate formula for the stint
of the modified Homer sefiog straight line, it will be
interesting to determine an approximate formula for the
start of the Horner sem-ilog straight line. Theoretically
speaking, the Homer semi30g straight line never yields
the correct straight hne. However, we can define an er-
ror criterion between the correct and computed Homer
slopes. Then, we. take. tie time at which the error criterion
is satisfied as the start of a Homer semfiog straight line.
As developed in Appendix B, m approximate formula
for the start of the Homer semilog straight line is given by
12ctPD –AtD[(rF +At)D] @ ‘btiD [–in p –27
1+-In 4+EXJ3AtD)+2S]- 1/AtD =0, . . . . . .. .(15)
where
an error criterion,
%., = slOpe of correct Horner semi-log straight
lie,
m~~~ = slope of computed Homer semilog stmaight
line, and
tpD = dimensiOIdess producing time.
For a given e (relative error), 6, skin, and producing time,
zeros of Eq. 15 with respect to AtD will give the sw
of the Homer semilog stmight line. It is interesting to
328 JoURNAL OF PETROLEUM TECHNOLOGY
—-—
observe that the start of the semilog Homer straight line
is a function of the”producing time, as expected.
A simple formula cannot be derived for the start of the
Homer semifog straight tie from Eq. 15 because it is
a transcendental function. Fig.. 4 presents values of9
15 as a” function” of dimensionless time for tPD = 10 ,
(3=10-4, S=0.0, and E= O.01. As can be seen from Fig.
4, E41. 15 has two roots (zeros) for the values of tPD, 6,
S, &d e given previously. The first zero resufts from the
early-time period, which can be seen easily from Fig. 2.
The second zero results from the radial inkinite-acting
period..l%e upper curve in Fig. 5 presents dimensionless
time for me start of the Homer semilog straight line as
a function of P for tPD = 10 6, S=0.0, and 6=0.1. As Cm
be seen from Eq. 15, the dimensiordess time for the stwt
of the Homer semilog S-might line is a very wesk func-
tion of skin. It is basically a function of the production
time and IS. This had been observed by Chen and
Binghamzz for tie constaht-wellbore-storage case. Eq.
15 slso can be used for the constant-we~bore-storage case
by substituting lICD for 13. However, UCD is a ve~
crpde approximation for “B.
Eq. 15 cm be quite useful for the design of buildup tests
for an optin@ value of a producingdme to achieve a cer-
tain accuracy for the Homer semilog straight line. Par-
ticularly, Eq. 15 will be usefuf for driflstem tests, since
production time is limited by production facilities.
For large producing tie, tPD, Eq. 15 can be simPlifi~
to
AfD=~. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. (16)
l+q. 16 also can be derived from drawdown solutions
by using the same principle given in Appendix B.
If p i.iapproximated by I/CD, Eq. 16 then can be writ-
ten as
AfD=:CD. . . . . . . . . . . . . . . . . . . . . . . . . . . ..(17)
As noted previously, Eq. 17 will yield very optimistic
values for tie stat of Homer semilog straight line if in-
deed the weflbore storage remains constant during the test.
The dimensionless time for the start of the motied
Homer semilog stmight line (details of derivations are
given in Appendm B) also can be given by
e[2@(tPAt)D +(tP+At)D] -b’2AtD2[(tP+AZ)Dl
.e-8~D(–1” &7-27+fII 4+2S)=0. .(18)
As in the Homer case, tie second root of Eq. 16 will give
tie dimensionless time for the sw of the modified Homer
semilog straight lime as a function of 8. Fig. 4 presents
vslues of Eq. 18 as a function of dimensionless time for
tPD=’106, “13=10 -4> S=0.0, wid 6=0.01.
The lower curve in Fig. 5 presents dimensionless time
for the start of the modified Horner semilog straight line
w a fUIItiWI of j3 for rPD=106, S=0.0, and c= O.01. The
sw of modified Homer semilog stmigbt we is also a
weak function of skin.
For large producing @ne, tPD, @. 18 cm 5iMP~i~ ~
AtD=:, . . . . . .. . . . . . . . . . . . . . . . . . . . . .. (19)
4@3
The modified Horner semilog straight line star@ at least
one-half cycle earfier than the Homer straight fine. fn
FEBRUARY 1985
other words, the time required for the stat of the modified
Homer stmight line is half of the time required for the
Homer straight line, as can be seen from Eqs. 16 and 19.
Depending on the formation and fluid parameters, hours
could be saved on the testing time.
Very simple qD and pD functions me used to explore
the effect of afterflow on the Horner analysis. It must be
recognized that the sandface flow rate (afterflow) has to
be measured to obtain accurate and reliable restdts from
the modified Homer analysis.
Even though the modified Horner anafysis improves the
semilog anafysis, it cannot be applied to very early-time
pressure data. The otier methods, such as continuous
mukirate, require prerequisitepD functions. Thus, decon-
volution methls will be used to obtain p,D functions (in-
fluence functions) and formation parameters in the
following section..
Deconvolution
Deteminin g wellbore geometries and reservok types
(fractured, layered, composite, etc.) is an important part
of well testing. The reservok engineer must have stiffi-
cient information about the system being analyzed. For
example, if w curves for fully penetrated wells ae used
for partially penetrated wells, both kh and damage skin
will be underestimated. Thus, the system identilcation
becomes a very important part of well testing. For in-
stance, identifying a one-half slope on a log-log plot of
tie pressure data will indicate. a vefically fractured well,
as two paral!el straight limes on a Horner graph wi13 in-
dicate a fractured resemoir. However, either wellbore
storage or’afterflow” usually dominates these chamcteristic
bebaviors of we.ffs and reservoirs durtig the early-time
period. Thus, the pressure behavior of formation without
the weflbore storage effect must be cahdated or the
wellbore storage effect on the formation pressure should
be minimized for the conventional identification of the
system. Tbis is merely the deconvolution of F@ 3a
tlqough 3C to calculate p,~(r~) from P WD (tD) and
qD (rD). seve~ graphs Of PsD(t~) VS. tD , such 2$ liim,
sphericsl, etc., will provide information about a given
wellbore geomet~ and reservoir.
This approach to system idendi5cation is not genersl,
but it uses our conventional knowledge about well and
resemoir behaviors. In general, the problem of the system
identification is much more complex because often we do
not fmow the governing differhisl equation.
It is worth repeating that the fluid flow in the forma-
tion is described by the linear dh%sivity equation for the
deconvolution methods given next.
There are several methods for tie deconvolution of Eqs.
3a through 3c. These metiods will be discussed in the
following section.
Lineariza tion of the Convolution Integral. In this sec-
tion, the “convolution equation,” Eq. 3c, will be solved
directly by using the limaiization niethod. Eq. 3C can be
discretized asn
PWZND”+l)= z ~“mlti%+l -~)i-o ~Dt
“ps~(~)d~. . . . . . . . . . . . . . . . . . . . ..(20)
329
il’’’’”F”’-’F‘“:”’’’””:;:L:L---- “-”-”
~’”:~,~l ,,!! ~~~~~~~~~
,o- ,0-’ , 0“
slut-l. n.,, At, hrs
Fig. 6—Calculated formation pressure drop using linearization
method and wellbore” pressure drop.
By using the trapezoid rule for integration in Eq. 20,
n
PwD(tDfI+l)= X bSD(tDt+I)Qi(tDn+I ‘tDi+l)
i=!l
+P.D(tDi)q~(2Dm+ I –2Di)l(tDi+I ‘2Di)/2. . . (21)
Eq. 21 givea a system of linear algebmic equations. The
coefficient inairiix of this system of equations is a lower
triangular niatri~ that is, all its nonzero elements aze in
tie lower right of the mat@ The system of equations
can be solved easily ‘by forward substitution.
For field data, p ~D sho~d be replaced by fp i –p wj)
and (p w, -PJ for drawduwn and b~dup tests, respec-
tively; and p,D should be replaced hy (pi –pw)f and
Q% –P&, ~bich we s@dface press~e +fferenc=without storage effecm (formation pressure drop). qD fur
drawdown and (1 –qD) for buildup must be used to
replace qD in Eq. 21 for ~eld case ch. If tie flOW ~S
radial, formation kh can be determined from the slupe of
the straight line if the (p ~, –pM)f vs. log’Af plot. Skin
factor also can be “determined from me conventiowf skin
formula. However, the trapezoidal method used in Eq.
21 gives osciffatory result.% As can be seen in Fig. 6, at
very early times, the cal~ulated values of @w. –P ~)f
oscillates. Furthermore, iugher-urder mefiods, inclu~g
the SiniPson rule, yield divergent results for the integral
in Eq. 20. ..” .’ .. .Hamminz23 sumzested a stible integration scheme for
evaluation ~f the ‘%wolntion integrul~. He also showed
that direct integration (discussed as the linearization
method) of convolution integmfs usually will r@t in w
owiflation.
The intigral in Eq. 20 can be approximated as
‘PSD(~Di+%)~1q~(2Dn+l-7)dT. . . . . . . ..(22)
r~i
The right side of Eq. 22 can he integrated directly.
Substitution of the integration resuhs in Eq. 20, and solv-
ing for p,D yields
p,D(tDn+,A)=~”~(f~+l)–s~ ,. .,.... . . . (23)
fZD(tDn+ I ‘tD.)
33U
,6,0-
=:
:,200
i . . . . . .“..,. !. ah”..!.
‘.”., !,. s,..,.!.
:
, m, -
+ ./”/., ./. .0,
/“: i“
--. -”.-.0 . .
,
10 ,o- ,0- 1,.’ d
SW-i- ‘m., .!, b
Fig. 7—Calculated formation pressure drop using Hamming
method and wellbore Pre?sure drop.
where “sum” is equal to
n—1
,2 psD(bi+fi)[qD(tDrI+l ‘b)
‘qD(tDn+I ‘tDi+I)l,
and the fist value ofp ,D is given by
PsD(tD%)=:;;;; — . . . . . . . . . . .. . . .. . . . . . ...(24)
As can be seen from Fig. 7, Eq. 23 gives a stable and
nonosdlatory integration scheme for the convolution in-
tegral given in Eq. 20. Furthermore, the derivative of the
sandface flow rate data, q~(tD), is not needed in Eq. 23.
However, for Eq. 21, q;(t) must be known. A iinite dif-
ference appromixation also can be used for q&D), but
some accufacy would be lost.
The semilog plot of @ ~, –p ,,.,+)f vs. At, given in Fig.
7 for radi+ synthetic buildup, test data, yields a straight
line starting kom a veiy early time (At=O.05 hours) @tb
a correct slope. The lower curve in the same figure is tbe
plot of (pW, –p;), which includes the we~bOre it~rage
effect.
Laplace Transform Decunvulntiun. The convolution of
RI. 4 yields
‘D
pSD(tD)= \ K(T)PWD(tD –7)d~, . . . . . . . . . . . . (25)~
where
“1
Mlr(z~)=c-1— . ................. (26)
s?D ($)
K(rD) can be cumputed eifher from the Laplace
transforms of qD(tD) data or a curve-fitted equation uf
qD(tD) data. However, it would be @e consuming to
invert all the gD(tD) data in Laplace space and transform
it back to real space in accordance with Eq. 26. Thus,
aPProhtion functions must be used for qD (tD) data.
Once an approximation is obtained, it will be easy m com-
pute K(tD) and integrate Eq. 25 to determine p,D(tD).
A few types of approximation functiuns can be used to
approximate qD (tD). We have tried rational functions,
power series, and ex~nentid functions. Expmentid func:
tions give a good representafiori of qD (tD) data. Ex-
JOURNAL OF PETROLEUM TECHNOLOGY
.—
—
TABLE 2—FRACTURED RESERVOIR ROCK
ANO FLUID DATA
B, bbl/STB 1.0
~, psi-l,..5
1,000
h,Dft 100
k,, md 400
pti, psia 4,216.94
q, STB/0 3>000
rw 0.35
R 10
tp, hours 10,000
k (spherical matrix used),.-5
~, Cp 0.8
+, 0.2
. 0.06
ponentiel functions approximation for qD can be
expressed as
qD(tD)=cleB1tD +c2e82tD +...cne8~tD, .:. (27)
where i= 1,2. .,.n.
After determining Ci xnd L3i from qD(tD) data, apply-
ing the Lapktce transformation to Eq. 27 and substituting
the resulting expression in Eq. 26 yields “..
1.”i?(y)= ~ , “ ..!:.’: ””””.”.:.:”’: ’(2. !)4
AS SugEeSM bv Sneddo”. 24 if the inverse tramfOrma-
tion d~~s not &fist, Eq. “28 should be divided by”p until
its inverse’ trxnsform is possible. For each ~vision ofp,
P~D(tD) fi Eq. 25 has to be differentiated.
We have seen from vw Everdngen’s7 and Hurst’sg
works thxt qD cti be approximated by m exponential
fonction as
qD=l_e-@D ,. ..,. . . . . . . . . . . . . .. . . . . . . (29)
which is the simplest form of Eq. 27. For this case, P$D
cm be wriiten directly from Eq. 3 as
mm=?’”’”;’‘+PwD(tD). . . . . . . . . . . (30)
TABLE 3—COMPARISON OF
COMPUTED AND ANALYTICAL
PRE?SURE FOR A FRACTURED
RESERVOIR
At
0.02
0.04
0.06
0.08
0.10
0.20
0.30
0,40
0.50
0.60
0.70
0.80
0.90
1.00
2.00
3,00
4.00
5.00
6.00
7.00
8.00
9.00
~131.93
133.57
134.66
136.47
136.12
138.27
139.62
140.62
141.42
142.10
142,68
143.20
143.66
144.08
146.94
148.!36
149.92
150.88
151.66
152.%
152.85
153.33
.-131.96
133.78
134.86
135.64
136.25
138.27
139.55
140.53
141.33
142.02
142.62
143.15
143.63
144;07
146.98
148,70
149.92
160.86
151.63
152.29
152.85
153.35
*65.33
98.32
115.30
124.37
129.44
137.03
138.87
139.83
140.85
141.59
142.24
142.81
142.32
143.78
146.84
148.60
149.85
150.80
151.59
152.24
152,81
153.32
From Eq. 30 it is very simple to calculate p,D if qD dat6
cm be approximated suc.csss~y ,by .,,using ,:.Eq. 29..
qD (tD ) data ds.o can be approximated by piecewise ex-
ponential fun~om for a selective interval for Eq. 26.
Curve-fit App:oximmtioiis’o fp,D”. p~~(tD) in Eqs. 3a
and 3:..c~wb&approximated by choosing suitable func-
tion~ approximations. These ~ctionel approximations
copld be power series, continued functions, rational func-
tions, or exponential functions. The success of this method
depends on how well the approximation function
represents the behavior of p,D. As noted earlier, the 10g
apPrOfimatiOn was used to approximate p,D (tD). In tis
case, the natural choice will be the power series of in tD
such tbatzz
m
“’””” >sD(tD)= x C@ I tD)i-l. . . . . . . . . . . . . . . ..(31){=1.. . . . . .—,
,.,
Substitution of Eq. 31 into Eq. 20 yields n number of
equations and “m number of unknown pxmneters,
.=(cI ,c2 .:.c~)T. In fact, obtxining these paramters, .;,
becomes an unconstrained opdmiiation problem. We have
appEed a fotirtb-degree polynomial, &f. 31, to weIfbore
pressure and rxte data (synthetic) that were obtained from
a fmctured reservoir for the resewoir and fluid pam.meteis
gjven in Table 2. The second column of Table 3 presents
the cxkdated values of Apti from curve-fit approxima-
tion for the fluid and reservok date given in Table 2, while
the third column of Table 3 presents the vxlues of Ap,f
calculated directly from q amlyticel solution without
storzge effect. The fourth column in Table 3 presents
&~, with the wellbore storage effect.
As seen from Table 3, dMerences in Ap,f from the
curve-fit approximation and the anxlydcel solutions are
xlmost identicel. The relative error decreases for lerge
times.
C0nc1u8i0nx
1. The convolution integml (superposition theorem) is
used for the analysis of continuously varying wellbore
flow rate and pressure. This analysis is ve~ similxr to
the conventional mukirxte meth0d3.
2. Wellbore storage (afterflow) effects can be present
to a significant degree in the Homer semi-log strxight Iiue.
The Horner an~ysis is modified by using measured sxnd-
face flow rate data to obtain a correct semilog straight
liie. The modified Horner semilog straight line steits”xt
last one-half cycle earfier tlmn the conventional Horner
FEBRUARY 1985 331
straight line. Approximate formulas afe presented for the
stat of the modified Homer and Homer semilog straight
limes as a function of sandface rate decline, production
time, and the relative error between the correct and com-
puted s30pes.
3. The formation pressure (influence function) can be
cakulated ffom the deconvolution of measured wellbore
pressure and sandface ffow rate data. Some new decon-
volution techniques are introduced to compute the format-
ion pressure (iiuence function) without wellbore storage
(atlertlow) effects. W&bore and reserv@r geometries ca
be identified from this computed fofmation pressure. Fur-
thei’more,, the conventional methods can be used to analyze
this computed formation pressure to detefmine formation
parameters.
4. Deconvolution of synthetic data from a homogeneous
and ffactured rekervoir shows tiiat it is possible to com-
pute the formation pressure from tbe”beghing of the test.
‘IW computed pressure reveak the characteristic behavior
of homogeneous and fractured reservoirs.
Acknowledgment
We thank Scblumberger We! Services for its peffnission
o pub3ish OdS paper aid acknowledge the en~u~gement
and help of Gerard Catala of Servic& Techniques
Scblumberger during the initial stage of this work. We
alio t@& H.J. Ramey Jr. for providing a solution for
the integral in Appendm A.
Nomencfatnre
B = oil fofmation volume factor, RBISTB
[res m3/stock tank m3]
ct = system total compressibility, psi-1
ma-l]
C = wellbore ,storage coefficient, bbllpsi
[m3/kPa]
CD = wellbore storage constant, dimensionless
h = formation tilckness, ft [m]
k = fofmation permeability, md
K = kernel of the convolution integraf
p: = deferential of pD
PDS = shut-in pressure drop, dimensionless
PD(tD) = fOfnJatiOn pressure, dimensionless
pi = initial pressure, psi [kPa]
PSD = PD +S = fofmation pressure includlng
skin, dmensi0n3ess
P WD = pressure drawdown, dmensicmless
PW = bOttotiole flowing pressure, psi [!-@a]
P~. = bottomhole shut-in pressure, psi ~]
q = stabflkid constant ate, STB/D
[stock-tank m3/d]
qD = s@face flow rate, dimensimdezs
qr = reference flow rate, B/D [m3/d]
qR = resewoti flow rate, BID [m3 /d]
q~ = smdface flow raw, B/IJ [~3/d]
– wellbore radius, ft [m]~w —
s = Laplme m~~fO~ vfiable
S = skin factor
t = time, hops
t~ = time, dimensionless
tp = production time, hours
332
tPD = dimensionless production t@e
T = transpose
At = running testing time, hours
ArD = shut-in time, dimensionless
a = O .000264kLWpc,r;
# = a positive constant
.y = 0.57722 . . . = Euler’s constant
p = viscosity, cp pa-s]
f = rate-preasufe convolved time function
~ = dwmny integration variable
+ = porosi~, fraction
- = LaPlace transform of
References
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APPENDIX A
Substitution of the exponential integral solution for pD
and Eq. 13 for qD in Eq. 12 yields
where
y= O.5772
and
Et(aAt)= \8M:du.
-m
Neglecting the imaginary term .i, the van Everdingen7
and Hursts forms are obtained as
&(AtD)= –%e-@MD [–III j3-2-y+fn 4
+Ei(@tD)]. . . . . . . . . . . . . . . . . . . . . ..(A-7)
[
1
pSD[(tP+&)D]’ ‘~~i ‘—
4(tP +At)D 1
.%E,(.&)-j””e-Bexp[ -&]0
dr.—+SqD(AtD), . . . . . . . . . . . .. . .. (A-l)
2(AtD -,)
~-pw(~)=~[l”g(%)+~(A’)l,-(A-8)
The values of integral :(&D) from f3kf. A-7 were com-
psred with the vafues computed from the Stehfest z
munericaf Laplace trsnsform inversion technique using
Eq. A-3. When AtDs 30, the difference between two
vslues of @tD) becomes less than 1%. The difference.
becomes smaller as AtD increases. Substitution of the log
approximations for the exponential integrafs given in
Eqs. A-I and A-7 for the integral yields (in pmcticd units)
where
~D=e —d==-LmrD. . . . . . . . . . . . . . . . . . . . ..(A-2)
Thus, tie two forms of gD wifl be used interchangeably.
The integral in Eq. A-1 cannot be integrated readily and,
unfortunately, van Everdingen7 and Ramey 10 dld not
present the &tds of integration. Ramey* provided me
a heurirtic derivation that wss edited out of his paper. 10
Rafney’s integration method is given below. The LapIace
transform of the integraf given in Eq. A-1 is
Z(S)= *KO(J). . .
The long-time approximation for KO(~
. . . . .. (A-3)
is
(vKoc~)=–lny+y. . . . . . . . . . . . . . . . ..(A4)
Substitution of Eq. A4 in Eq. A-3 yields
1j(s) = –— (h s–in 4+2T). . . . . . . . . ..(A-5)
2(s+.0)
The inverse Laplace transform of Eq. A-5 yields
@ro)= –fie-@tiD [–in &~i-2y+fn 4
+Ei(6AtD)], . . . . . . . . . . . . . . . . . . . . ..(A-6)
‘Ranw H.J, Jr.: P+rsmal eommunkallon S$anfwd U., Stanford, CA (Mm+l 1984).
FEBRUARY 19S5
where
i(At)=*e ‘aAr[–fn f3-27+fn 4+ Ei(aAf)
+~. . . . . . . . ... . . . . . . . . . . . . . . . . . . . ..(A-9)
The long-time approximation for g(At) maybe obtained
by substituting limiting fbims of ci+tain terms in Eq.’ A-7
for long times. For lsrger values of At, the term
e ‘@(-bI j3-27+ln 4+2$ in Eq. A-9 approaches
zero. The term e ‘aNEi(aAt) can be approxinsted as
e-a&Ei(aAt)=~, . . . . . . . . . . . . . . . . . ..(A-1O)
when &+l.
Substituting Eq. A-10 in Eq. A-9 yields
~(At) =~ (A-11)2.3026dr” ““”””’”””””’’”””””””
Further substitution of Eq. A-11 in Eq. A-8 gives Eq. 14
in the main text.
APPENDIX B
The dimensionless form of Eqs. A-8 and A-9 cm be writ-
ten as
PD(A’)=05[”[-1+’(A’D)]w333
where
~(AtD)= %. ‘~ND [–In (3-2y+ln 4+ Ei@AtD)
+2,g. . . . . . . .. . . . . . . . . . . . . . . . . . ...03-2)
The differentiation of Eq. B-1 with respect to
In{[(tp +At)D]/AfD} yields
‘com=0.5++[g(Ad],....... ...........@-3)
where. mcom is the computed dimensionless slope
of the Homer semifog straight line and x equals
ln[(f,n +At)D/AtD] .
Let us &fine a relative error for tie computed slope as
~= I =Ij ...... .... .........
where
mcom> 0.5.
Substitution of Eq. BzI in Eq. B-3 yields
. (B-I)
df(AtD)~= ~~—.. . .. . . . . ... .. . . . . . . . . . . . . . . . . . (B-5)
Ax
Taking the &rivative of $(AtD) with respect to x and
substituting in Eq. B-5 yields
AtD[(tP +M)DI~=
[
A3e-~&D [–1. 13-2-y+ln 4
2tpD
.1+.Ei((3AtD)+2S]-~ . . . . . . . . . ...@-@
Eq. B-5 gives “tie dimensionless dme for the start of the
Homer semilog straight line as a function of e, 0, fPD,
and S.
A formula for the dimensionless time for the start of
the modified Homer semilog straight line can be found
from Eqs. A-1 and A-2 as follows. Let us rewrite Eq.
A-1 in terms of the modified Homer time
f’D@@=”4w%31”1
+— I +%$(AtD), . .
26AtD
,. (B-7)
where
$(AtD)=!4e-5ti.c (–fn fl-2y+ln 4+2S). .: .(B-S)
lA$(AtD) in Eq. A-7 is the ordy remaining term that is
not included in the modified Horner time. Thus, the
relative error of slope of the computed modified Homer
semilog straight line (as in the Homer case) can be ex-
pressed as
df(AtD)~. —...- . . . . . . .. . . . .. . . . . . . . . . . . .
Ax
(E-9)
where
‘=1”[(%%]++ ‘@-’o)
Differentiation of Eq. B-8 with respect to x and substhw
tion of the result in Eq. B-9 yields
.(–in 6–2y+ln 4+2S). . . . . . . . . . . . .(B-11)
Eq. B-11 gives the dimensionless time for the start of the
modified Homer semiIog stiaigbt line as a fiction of e,
o, tPD, and skin.
S1 Metric Conversion Factors
bbl X 1.589873 E–01 = m3
Cp x 1 .“* E–03 = Pas
ft X 3.048* E–01 = m
psi x 6.894757 E+OO = kPa
.Gmversl.m Iaclor is exact JP’f
0ri@n=4 rmnusuiDt rec.stved in the Satiety of Pe,,%um Engineers ollb Od, 5, ? 9S3.
Pew accaPted for P.MWO. M.,ch 11. I ss,. ~~vi,ed manuscript ,ece~ved ‘e*
14, 19%. PaPer (SPE 121771 first Prem.ted at the 19S3 SPE Annual T@ch”icnl Con.
term. and mhlbitlo. held in S.. Franckm 0~. ~s.
334 JOURNAL OF PETROLEUM ~CHNOLOGY
4aoo.o
4280.0
4970.0
4ZSOm0
4260.0
4240,0
4aao,o
4220,0
4210s0
4aoo.o
lb’HORNER & MODIFIED
Fig, 3-Modlfled Horner and Horner plots
\
Y‘\
LEQENDhornermo e_ ,dKd l!w!!,e~
\\
\
\.
—.
.
—
—
\
‘\
decllne sandfaco rate,
I
—
—
.
T
—
)2/77
1800.0
1700,0
1600,0
1600,0
1400.0
Iaoo.o
1200s0
1100.0
1000.0
IIII
TTll
.
1
.
Ill
111
P
.
“
LEGENDG wdlbOre DKM3SUWi
_m!M?lm.EE !E!M!!u0
Fig, 4-Calculated formation pressuredropusingIkwkatkmmethod and wellbore pressure drop,
1’
1A/7,7
1600,0
?Emh n
1400,0
1s00.0
1200,0
1100,0
[ m II’
100
Fig,
0—
0—
/
/
.01--110-8
5-Calculated formation pressure drop
TIME, HOUR8
using Hamming method and
10’
wellbore pressure
II
drop,
10R
aooooo
2800.0
2600.0
z S400.Oa
w“eg 2200,0
$
aan
z2000.0
Tt=sx@
1600.0
~
dwa
1600.0
1400.0
1200.0
1000.Q
——
——
LEQENDo formation pre8sun3
LY?s!!!l ,ore ixessurp_
—
—
—
—
—
—
—
—
Ii)-’
—
>
—
—
—
—
—
—
.
r
.
.
i)
.
@@”
—-
—
—
6
—
—
—
—
—
—
-
I
Fig, 6-Calculated formation pressure drop using polynomial approximation and welibore pressure for afractured reservoir,